I would cut back dead wood now, to just above a healthy bud of leaf and yes, I would also cut out crossing and rubbing branches so it grows with an open canopy. It will fill in the gaps with new stems and plenty of foliage but be a healthier tree. In later years you can remove lower branches to just above head height for passers by.

"We don't stop playing because we grow old; we grow old because we stop playing." - George Bernard Shaw

When you cut dead wood you need to go back to live wood or you risk leaving infected or damaged cells that cause die-back to continue down the stems. Best done in autumn after leaf fall when you can see the shape of the tree more easily or in late spring/early summer when sap flow has slowed.

"We don't stop playing because we grow old; we grow old because we stop playing." - George Bernard Shaw